Problem: Simplify; express your answer in exponential form. Assume $a\neq 0, r\neq 0$. $\dfrac{{(a^{5}r^{4})^{-4}}}{{(ar^{5})^{-3}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{5}r^{4})^{-4} = (a^{5})^{-4}(r^{4})^{-4}}$ On the left, we have ${a^{5}}$ to the exponent ${-4}$ . Now ${5 \times -4 = -20}$ , so ${(a^{5})^{-4} = a^{-20}}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{5}r^{4})^{-4}}}{{(ar^{5})^{-3}}} = \dfrac{{a^{-20}r^{-16}}}{{a^{-3}r^{-15}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{-20}r^{-16}}}{{a^{-3}r^{-15}}} = \dfrac{{a^{-20}}}{{a^{-3}}} \cdot \dfrac{{r^{-16}}}{{r^{-15}}} = a^{{-20} - {(-3)}} \cdot r^{{-16} - {(-15)}} = a^{-17}r^{-1}$